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Chapter 5: Problem 6

Find the amount (future value) of each ordinary annuity. \(\$ 150 /\) month for 15 yr at \(10 \%\) /year compounded monthly

Short Answer

Expert verified
The future value of the annuity is approximately \(\$44,197.93\).

Step by step solution

01

Convert the annual interest rate to a monthly interest rate

Since the interest rate is compounded monthly, we need to convert the annual interest rate to a monthly interest rate. To do this, divide the annual interest rate by the number of compounding periods per year (in this case, 12 months): \[i = \frac{0.10}{12}\]
02

Determine the total number of periods

The total number of periods is the product of the number of years and the number of compounding periods per year: \[n = 15 \text{ years} * 12 \text{ months/year}\]
03

Apply the future value formula

Now that we have \(i\) and \(n\), we can apply the future value formula to find the future value of the annuity: \[FV = 150 * \frac{(1 + i)^n -1}{i}\] Substitute the values of \(i\) and \(n\) and solve for the future value of the annuity.
04

Calculate the future value of the annuity

Using the values of \(i\) and \(n\), we have: \[FV = 150 * \frac{(1 + \frac{0.10}{12})^{15 * 12} -1}{\frac{0.10}{12}}\] Now, calculate the future value of the annuity. The future value of the annuity is approximately \(\$44,197.93\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordinary Annuity
An ordinary annuity involves regular payments or deposits made at the end of each period, like when you save a fixed amount monthly in a savings account or invest in a financial product. For the example problem, this involves depositing $150 each month for 15 years. These annuities are contrasted with annuities due, where payments are made at the beginning of each period.

Ordinary annuities are commonly used in calculating loan payments, contributions to retirement savings, or any scenario where consistent payments are made over time. Understanding how ordinary annuities work can help you plan for and manage your finances effectively.
Compounded Interest
Compounded interest refers to the process of earning interest on both the initial principal and the accumulated interest from previous periods. It means that you'll earn interest on your interest, which can significantly increase the amount in your account over time.

In the provided exercise, the interest is compounded monthly, meaning every month the interest amount is calculated and added to the principal. This has a multiplying effect on the total future value of the annuity compared to simple interest, where only the principal earns interest every time.
Monthly Compounding
Monthly compounding indicates that the interest on an investment is calculated and applied to the balance twelve times a year. This frequency allows the annuity to generate more interest buildup compared to annual compounding. With each compounding period, the principle amount grows, which multiplies further during the next period.

To calculate the future value in the situation described, the annual interest rate is divided by 12 to find the effective monthly interest rate. This conversion is essential to ensure accurate calculations and reflections of how compounding occurs in shorter periods.
Annuity Formula
The annuity future value formula calculates how much money will be accumulated by the end of the annuity鈥檚 term, based on regular, periodic payments. The formula used is:\[FV = PMT * \left( \frac{(1 + i)^n - 1}{i} \right)\]where:
  • \(FV\) is the future value of the annuity.
  • \(PMT\) is the periodic payment amount ($150 in this case).
  • \(i\) is the periodic interest rate (annual rate converted to a monthly rate).
  • \(n\) is the total number of payment periods (15 years times 12 months).

By entering the respective values into the formula, you can calculate how much the annuity will grow over time. This formula is a powerful tool for predicting future savings and planning financial goals, allowing individuals to understand how their regular investments will compound and grow.

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Most popular questions from this chapter

IRAs Martin has deposited \(\$ 375\) in his IRA at the end of each quarter for the past 20 yr. His investment has earned interest at the rate of \(8 \% /\) year compounded quarterly over this period. Now, at age 60 , he is considering retirement. What quarterly payment will he receive over the next 15 yr? (Assume that the money is earning interest at the same rate and that payments are made at the end of each quarter.) If he continues working and makes quarterly payments of the same amount in his IRA until age 65, what quarterly payment will he receive from his fund upon retirement over the following \(10 \mathrm{yr}\) ?

Use logarithms to solve each problem. How long will it take \(\$ 5000\) to grow to \(\$ 6500\) if the investment earns interest at the rate of \(12 \% /\) year compounded monthly?

SINKING FuNDS The management of Gibraltar Brokerage Services anticipates a capital expenditure of \(\$ 20,000\) in 3 yr for the purchase of new computers and has decided to set up a sinking fund to finance this purchase. If the fund eams interest at the rate of \(10 \% / y e a r\) compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.

Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of \(9 \% /\) year compounded monthly. If the future value of the annuity after 10 yr is \(\$ 60,000\), what was the size of each payment?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The future value of an annuity can be found by adding together all the payments that are paid into the account.
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